Table Of ContentPURE AND APPLIED MATHEMATICS
AWiley-InterscienceSeriesofTexts,Monographs,andTracts
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SEMIGROUPS AND
COMBINATORIAL
APPLICATIONS
SEMIGRUUPS AND
COMBINATORIAL
APPLICATIONS
GERARD LALLEMENT
PennsylvaniaState University
.A WILEY-INTERSCIENCE PUBLICATION
JOHN WILEY & SONS, New York - Chichester - Brisbane - Toronto
Copyright© 1979byJohnWiley&Sons,Inc.
Allrightsreserved. PublishedsimultaneouslyinCanada.
Reproductionortranslationofanypartofthiswork
beyondthatpermittedbySections 107or 108ofthe
1976UnitedStatesCopyrightActwithoutthepermission
ofthecopyrightownerisunlawful. Requestsfor
permissionorfurtherinformationshouldbeaddressedto
thePermissionsDepartment,JohnWiley&Sons,Inc.
LibraryofCongressCataloginginPublicationData:
Lallement,Gerard, 1935—
Semigroupsandcombinatorialapplications
(Pureandappliedmathematics)
“AWiley-Interscienccpublication.”
Includesbibliographicalreferencesandindex.
1. Semigroups. 2. Machinetheory. 3. Combina-
torialanalysis. I. Title.
QAl7l.L27 512’.2 78-23561
ISBN0-471—04379-6
PrintedintheUnitedStatesofAmerica
10987654321
PREFACE
Thepurposeofthisbookistopresentthosepartsofthetheoryofsemigroups
that are directly related to automata theory, algebraic linguistics, and com-
binatorics. In the last 20 years a growing number ofpublications in these
mathematical disciplines contained results and methods pertaining to the
algebraic theory of semigroups. This has contributed to a considerable
enrichment ofthistheory, enlargedits scope, and improved itspotential for
becoming a major domain ofalgebra. In return, it appears that semigroup
theoryprovides aconvenientgeneralframeworkforunifyingand clarifying
a number oftopics in fields that seem, at first sight, unrelated.
The book is intended as atextbookforgraduate students in mathematics
and computer science and as a reference book for researchers interested in
associative structures. For the first halfofit (Chapters 1 to 4) the only pre-
requisite is a rudimentary knowledge ofelementary algebra (definition ofa
group, ring, field, lattice, etc.). In the second half, which analyzes the inter-
play between semigroups, automata, languages, and combinatorics, it often
has been necessary to call upon other branches ofmathematics (e.g. logic,
probability theory). In such instances, motivated by the desire of keeping
thepresentationwithinreasonablelimits,wegivetheappropriatedefinitions
or state the results in the form needed in the text, and indicate the proper
references for further detail. At the end ofeach chapter the reader will find
a number of exercises with references for solutions, as well as additional
reading references and historical comments.
Chapters 1 to4andtheportionofChapter5coveringfreesemigroupsand
monoids contain the basic concepts in the theory of algebraic semigroups
(Green’s relations, the Suschkewitsch—Rees theorem, the prime decomposi-
tion theorem for transformation semigroups). Free semigroups, languages,
and codes are introduced in Chapter 5, where we also briefly discuss decid-
‘ability and algorithms, since these are recurring themes in the applications.
InChapter6wepresenttheelementsofautomatatheory, includingKleene’s
theorem, and the recent variety theorem ofEilenberg. The emphasis in this
V
vi PREFACE
chapterisontransformationmonoidsandsyntacticmonoidsofrecognizable
languages. The star operation on languages and how it affects syntactic
monoidsaretheobjectivesofChapter7.Itisshown,inparticular,thatdeeper
results on languages expressible with the use ofthe star operation depend
crucially on the study oflanguages generated by prefix codes. This study is
the purpose of Chapter 8, where we present the basic results in an area
closelyrelatedtopermutationgroupsandcombinatorics.Passingtocontext-
free languages (called here algebraic languages) in Chapter 9, we outline
their most immediate properties, show their relationship with (algebraic)
power-seriesinnoncommutingvariables,andspecifythepositionofrational
languages in this larger class. In the final two chapters we explore further
connections between semigroups and combinatorial questions related
successively to the Burnside problem and to MacMahon’s master theorem.
This book does not pretend to cover all of the existing applications of
semigroup theory. I have selected a number oftopics with two related im-
peratives in mind. First, I wanted to emphasize the generic aspect ofsemi-
group theory, since its specific aspect (i.e. the study ofsemigroups per se)
has already been presented in several books [6.g. the two volumes ofA. H.
Clifford and G. B. Preston (1961, 1967)]. Second, I wanted to show ex-
plicitly how the study of semigroups, viewed as members of a generic
structure, inserts itselfinto the study ofmore complex branches ofmath-
ematics like automata theory, the theory of formal languages, and com-
binatorics in general.
Most ofthe materialpresented here emerged slowlyfrom several second-
year graduatecourses I gave to students ofthe Mathematics and Computer
ScienceDepartmentsatthePennsylvaniaStateUniversityduringtheperiod
1970—1975, and also fromnumerous talks inflicted upon my colleagues and
studentsinvariousseminarsduringthesameperiodoftime. Ihavebenefited
greatly from many ofmylisteners’ criticisms and suggestions, and I express
mygratitude to all ofthem.
During the academic year 1975—1976 J. F. Perrot invited me to become a
member of a team of lecturers in a third cycle course at the University
Pierre and Marie Curie (Paris VI). A large number oftopics contained in
Chapters2, 3, and 5 to 9werecoveredinthiscourse. Thisprovidedmewith
the best possible opportunity for completing the final draft of this book,
and I wish to thank J. F. Perrot, J. Berstel, D. Perrin, and J. Sakarovitch
for contributing to many valuable discussions and making many useful
suggestions in a very friendly atmosphere.
My former student M. Keenan meticulously read the entire manuscript,
and I thank him deeplyfor his constanthelp in improvingthe presentation.
I am also most grateful to J. F. Perrot, D. Perrin, and G. Jacob for their
numerouscomments and suggestionsconcerning Chapters 7, 8, and 10.
PREFACE vii
M. P. Schiitzenberger introduced me to automata, languages, and com-
binatorics almost 15 years ago. This book, in which I attempt to give a
reflection ofhis pioneering work and ofthe stimulating effect ofhis ideas,
is a small token ofmy gratitude and personal indebtedness.
GERARD LALLEMENT
UniversityPark, Pennsylvania
January1979
